Structure of a Morse form foliation on a closed surface in terms of genus. (English) Zbl 1223.57022

Gelbukh, Irina

Differ. Geom. Appl. 29, No. 4, 473-492 (2011).

Let \(\omega\) be a closed \(1\)-form with Morse singularities on a genus \(g\) closed surface \(M\). The form \(\omega\) defines a foliation \(\mathcal{F} _{\omega}\) on \(M\backslash Sing\;\omega\) and a singular foliation \(\overline{\mathcal{F}}_{\omega}\) on the whole \(M\). A leaf \(\gamma \in\mathcal{F}_{\omega}\) is compactifiable if \(\gamma\cup Sing\;\omega\) is compact. Non-compactifiable leaves of \(\mathcal{F}_{\omega}\) form minimal components \(\mathcal{C}_{j}^{\min};\) each such leaf is dense in its minimal component. If \(\gamma\) is a compact singular leaf, a small closed tubular neighborhood of \(\gamma\) is denoted by \(V(\gamma)\).
The main result of this work is the relation \(c(\omega)+\sum_{\gamma}g(V(\gamma))+g(\bigcup \overline{\mathcal{C}_{j}^{\min}})=g,\) where \(c(\omega)\) is the number of homologically independent compact leaves. This result allows the author to prove a criterion for compactness of \(\overline{\mathcal{F}}_{\omega}\), to estimate the number of its minimal components and to give an upper bound on the rank \(rk\;\omega,\) in terms of genus.

Reviewer: Charalampos Charitos (Athens)

MSC:

57R30	Foliations in differential topology; geometric theory
58K65	Topological invariants on manifolds

Keywords:

Morse form foliation; minimal component; compact singular leaves; genus; isotropic subgroup

References:

[1]	Aranson, S. K.; Zhuzhoma, E. V., On the structure of quasiminimal sets of foliations on surfaces, Russ. Acad. Sci. Sb. Math., 82, 397-424 (1995) · Zbl 0842.57024
[2]	Arnoux, P.; Levitt, G., Sur lʼunique ergodicité des 1-formes fermées singulières, Invent. Math., 84, 141-156 (1986) · Zbl 0577.58021
[3]	Farber, M., Topology of Closed One-Forms, Math. Surv., vol. 108 (2004), AMS · Zbl 1052.58016
[4]	Gelbukh, I., Presence of minimal components in a Morse form foliation, Differ. Geom. Appl., 22, 189-198 (2005) · Zbl 1070.57016
[5]	Gelbukh, I., Number of minimal components and homologically independent compact leaves for a Morse form foliation, Stud. Sci. Math. Hung., 46, 547-557 (2009) · Zbl 1274.57005
[6]	Gelbukh, I., On the structure of a Morse form foliation, Czech. Math. J., 59, 207-220 (2009) · Zbl 1224.57010
[7]	I. Gelbukh, The number of minimal components and homologically independent compact leaves of a weakly generic Morse form on a closed surface, Rocky Mt. J. Math., in press.; I. Gelbukh, The number of minimal components and homologically independent compact leaves of a weakly generic Morse form on a closed surface, Rocky Mt. J. Math., in press. · Zbl 1280.57021
[8]	Harary, F., Graph Theory (1994), Addison-Wesley Publ. Comp.: Addison-Wesley Publ. Comp. Reading, MA · Zbl 0797.05064
[9]	Imanishi, H., On codimension one foliations defined by closed one forms with singularities, J. Math. Kyoto Univ., 19, 285-291 (1979) · Zbl 0417.57010
[10]	Jiménez López, V.; Soler López, G., Transitive flows on manifolds, Rev. Mat. Iberoamericana, 20, 107-130 (2004) · Zbl 1063.54032
[11]	Kono, S., The structure of quasiminimal sets, Proc. Japan Acad., 46, 599-604 (1970) · Zbl 0219.54041
[12]	Levitt, G., Feuilletages des surfaces, Ann. Inst. Fourier, 32, 179-217 (1982) · Zbl 0454.57015
[13]	Levitt, G., 1-formes fermées singulières et groupe fondamental, Invent. Math., 88, 635-667 (1987) · Zbl 0594.57014
[14]	Maier, A. G., Trajectories on closed orientable surfaces, Math. Sb., 12, 71-84 (1943) · Zbl 0063.03856
[15]	Melʼnikova, I., An indicator of the noncompactness of a foliation on \(M_g^2\), Math. Notes, 53, 356-358 (1993) · Zbl 0809.57018
[16]	Melʼnikova, I., A test for non-compactness of the foliation of a Morse form, Russ. Math. Surveys, 50, 444-445 (1995) · Zbl 0859.58005
[17]	Melʼnikova, I., Maximal isotropic subspaces of skew-symmetric bilinear mapping, Mosc. Univ. Math. Bull., 54, 1-3 (1999) · Zbl 0957.57018
[18]	Zorich, A., Hamiltonian Flows on Multivalued Hamiltonians on Closed Orientable Surfaces (1994), Max-Planck Institut für Mathematik: Max-Planck Institut für Mathematik Bonn